The Algebraical Braid Group
The Algebraical Braid Group
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代数辫群
DOI：
10.2307/1969219
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发表时间：
1947
影响因子：
4.9
通讯作者：
F. Bohnenblust
中科院分区：
文献类型：
--
作者：
F. Bohnenblust
在他的论文'Theorie der Z6 pfe' E. Artin'提出了一个理论的辫子的基础上研究他们的投影在一个二维平面。在投影中，每一股辫子都表现为一条线，通常是垂直的，但在某些水平上，两条相邻的辫子会互换位置，一条辫子在另一条辫子的前面交叉。位置i处的链在位置（i + 1）处的链之前的交叉由表示。如果后者在前者前面交叉，则交叉用ai表示。编织物完全被描述为a的幂积。对于I i j I > 2，辫子aiaj和ajai显然在几何上是等效的，并且对于辫子aiai+iai和ai+1 iOiaO也是如此。这些等价是辫群的定义关系。换句话说，这个群是通过引入关系aiaj / ajai（Ii-j i > 2）和aiaiojai ai ai+1aiai从由a生成的自由群导出的。这些结果在上述Artin的论文中得到了充分的承认，并特别导致了辫子的分类。然而，这些证明是部分直观的。在最近的一份文件2阿廷回到这个问题，利用一个更直接的方法，允许一个优雅的和完全严格的处理问题。而不是利用投影，辫子坐标被引入。a在理论的后期才出现，但即使如此，它们也只起着次要的作用。由其性质的问题是几何和整个主要部分，他的文件阿丁广泛使用几何考虑。在本文中，一个纯粹的群论问题被认为是。“代数辫群”被定义为由具有上述关系的a生成的群。这组分析代数的方法和这种分析导致相同的基本结果所取得的阿丁;第一个标准平等的两个要素的群体，其次是一个真正的代表性的群体作为一组取代在一个自由的群体。通过结合几何理论与代数方法，它最终表明，在a的定义关系是几何辫子群的定义关系。
In his paper 'Theorie der Z6pfe' E. Artin' presented a theory of braids based on a study of their projections on a two-dimensional plane. In the projection each strand of a braid appears as a line, vertical in general, but at certain levels two neighboring strands interchange position, one strand crossing in front of the other one. The crossing of the strand in position i in front of the strand in position (i + 1) is denoted by as . If the latter crosses in front of the former the crossing is denoted by ai A braid is completely described as a power product in the a's. The braids aiaj and ajai for I i j I > 2 are obviously geometrically equivalent and the same holds for the braids aiai+iai and ai+1iOiaO . These equivalences are defining relations for the braid group. In other words, this group is derived from the free group generated by the a's by introducing the relations aiaj / ajai ( I i-j i > 2) and aiaiojai ai+1aiai . These results were fully recognized in the paper of Artin mentioned above and led in particular to a classification of braids. The proofs, however, are partially intuitive. In a recent paper2 Artin returned to this question making use of a more direct approach which permits an elegant and completely rigorous treatment of the problem. Instead of utilizing the projection, braid coordinates are introduced. The a's appear at a late stage in the theory but even then they play only a minor role. By its nature the problem is geometrical and throughout the major part of his paper Artin makes extensive use of geometrical considerations. In the present paper a purely group-theoretical problem is considered. The 'algebraical braid group' is defined as the group generated by the a's with the relations mentioned above. This group is analyzed by algebraical methods and this analysis leads to the same fundamental results obtained by Artin; first a criterion for the equality of two elements of the group, secondly a true representation of the group as a group of substitutions in a free group. By combining the geometrical theory with the algebraical approach it is finally shown that the defining relations in the a's are defining relations for the geometrical braid group.